On 5/26/2012 12:11 PM, Bruno Marchal wrote:
On 26 May 2012, at 17:56, meekerdb wrote:
On 5/26/2012 2:16 AM, Bruno Marchal wrote:
On 02 Mar 2012, at 06:18, meekerdb wrote (two month agao):
On 3/1/2012 7:37 PM, Richard Ruquist wrote:
Excerpt: "Any system with finite information content that is consistent can be
formalized into an axiomatic system, for example by using one axiom to assert the
truth of each independent piece of information. Thus, assuming that our reality has
finite information content, there must be an axiomatic system that is
isomorphic to our reality, where every true thing about reality can be proved as a
theorem from the axioms of that system"
Doesn't this thinking contradict Goedel's Incompleteness theorem for consistent
systems because there are true things about consistent systems that cannot be
derived from its axioms? Richard
Presumably those true things would not be 'real'. Only provable things would be true
of reality.
Provable depends on the theory. If the theory is unsound, what it proves might well be
false.
And if you trust the theory, then you know that "the theory is consistent" is true,
yet the theory itself cannot prove it, so reality is larger that what you can prove in
that theory.
So in any case truth is larger than the theory. Even when truth is restricted to
arithmetical propositions. Notably because the statement "the theory is consistent"
can be translated into an arithmetical proposition.
Bruno
Does arithmetic have 'finite information content'? Is the axiom of succession just one
or is it a schema of infinitely many axioms?
Arithmetical truth has infinite information content.
That's what I thought. So the above Excerpt does not contradict Godel's incompleteness
because it refers to "systems with finite information content".
Peano Arithmetic has about 5K of information content,
Which is just the information in the axioms (actually that number seems high to
me).
Brent
even with the infinitely many induction axioms, for they are simple to generate. There
are two succession axioms (0 ≠ s(x), and s(x) = s(y) .-> x = y. Those are not scheme of
axioms.
Bruno
http://iridia.ulb.ac.be/~marchal/
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